2023 AMC 12A Problems/Problem 1
- The following problem is from both the 2023 AMC 10A #1 and 2023 AMC 12A #1, so both problems redirect to this page.
Contents
- 1 Problem 1
- 2 Solution 1
- 3 Solution 2
- 4 Solution 3
- 5 Solution 4
- 6 Solution 5 (Under 20 seconds)
- 7 Solution 6 (simple linear equations)
- 8 Solution 7
- 9 Solution 8
- 10 Video Solution by Little Fermat
- 11 Video Solution by Math-X (First understand the problem!!!)
- 12 Video Solution by Power Solve (clear)
- 13 Video Solution 1 (⚡Under 1 min⚡)
- 14 Video Solution by CosineMethod [🔥Fast and Easy🔥]
- 15 Video Solution
- 16 Video Solution (easy to digest) by Power Solve
- 17 Video Solution by DR.GOOGLE (YT: Pablo's Math)
- 18 See also
Problem 1
Cities and are miles apart. Alicia lives in and Beth lives in . Alicia bikes towards at 18 miles per hour. Leaving at the same time, Beth bikes toward at 12 miles per hour. How many miles from City will they be when they meet?
Solution 1
This is a problem, so let be the time it takes to meet. We can write the following equation: Solving gives us . The is Alicia so
~zhenghua
Solution 2
The relative speed of the two is , so hours would be required to travel miles. , so
~walmartbrian
Solution 3
Since mph is times mph, Alicia will travel times as far as Beth. If is the distance Beth travels, Since this is the amount Beth traveled, the amount that Alicia traveled was
~daniel luo
Solution 4
Alice and Barbara close in on each other at 30mph. Since they are 45 miles apart, they will meet in t = d/s = 45miles / 30mph = 3/2 hours. We can either calculate the distance Alice travels at 18mph or the distance Barbara travels at 12mph; since we want the distance from Alice, we go with the former. Alice (and Barbara) will meet in 1 1/2 hours at 18mph x 3/2 hours = 27 miles from A.
~Dilip
Solution 5 (Under 20 seconds)
We know that Alice approaches Beth at mph and Beth approaches Alice at mph. If we consider that if Alice moves miles at the same time Beth moves miles —> miles left. Alice then moves more miles at the same time as Beth moves more miles. Alice has moved miles from point A at the same time that Beth has moved miles from point B, meaning that Alice and Beth meet miles from point A.
~uwusugardaddy14
Solution 6 (simple linear equations)
We know that Beth starts 45 miles away from City A, let’s create two equations: Alice-> Beth-> [-12 is the slope; 45 is the y-intercept]
Solve the system:
So,
Solution 7
Since Alicia and Beth's speeds are constant, they are directly proportional to their distances covered, so the ratio of their speeds is equal to the ratio of their covered distances. Since Alicia travels of their combined speed, she travels of the total distance.
-Benedict T (countmath1)
Solution 8
Note that Alicia and Beth must have travelled 45 miles together in order for them to meet together. While travelling the 45 miles, Alicia came closer towards Beth and Beth came closer towards Alicia. Since Alicia is travelling faster than Beth, we know that they will meet slightly closer towards city B from the middle. Also, the distance remaining between Alicia and Beth as they bike toward each other is proportional to their combined velocity. The combined velocity is . The time it takes to travel 45 miles going at 30 miles per hour can be calculated using simple algebra.
Let be the time it takes for Alicia and Beth to meet together.
\begin{align*} 18t + 12t = 45 \\ 30t = 45 \\ t = \frac{3}{2} \end{align*}
It takes hours of cycling by both Alicia and Beth at their respective biking speeds in order to meet together. Also, note that the question asks for the distance from City A they will be when they meet. This is the same as the distance travelled by Alicia from City A. Alicia travels at 18 mph, which turns into a distance = speed x time question.
\begin{align*} d &= st \\ d &= 18 \times \frac{3}{2} \\ d &= 27 \end{align*} The distance from City A when Alicia and Beth meet is .
Note - Even though it might seem like there is a lot of work for this problem, it would typically take a very short time to think this entire process out.
~S. Khunti
Video Solution by Little Fermat
https://youtu.be/h2Pf2hvF1wE?si=BOVppjT8sSTp2m0V&t=118 ~little-fermat
Video Solution by Math-X (First understand the problem!!!)
https://youtu.be/GP-DYudh5qU?si=TdTUYY4j-g_K_8-b&t=115 ~Math-X
Video Solution by Power Solve (clear)
https://youtu.be/YXIH3UbLqK8?si=YFt4h7wm7XHun1lK
Video Solution 1 (⚡Under 1 min⚡)
~Education, the Study of Everything
Video Solution by CosineMethod [🔥Fast and Easy🔥]
https://www.youtube.com/watch?v=WkRt_n1TEKY
Video Solution
~Steven Chen (Professor Chen Education Palace, www.professorchenedu.com)
Video Solution (easy to digest) by Power Solve
https://www.youtube.com/watch?v=8huvzWTtgaU
Video Solution by DR.GOOGLE (YT: Pablo's Math)
See also
2023 AMC 10A (Problems • Answer Key • Resources) | ||
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All AMC 10 Problems and Solutions |
2023 AMC 12A (Problems • Answer Key • Resources) | |
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Followed by Problem 2 |
1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 • 16 • 17 • 18 • 19 • 20 • 21 • 22 • 23 • 24 • 25 | |
All AMC 12 Problems and Solutions |
The problems on this page are copyrighted by the Mathematical Association of America's American Mathematics Competitions.